REFRACTION PHENOMENA 495

become distinctly separated, as per the accompanying table1 for ju = 1.31, in which h = solar elevation, A0 = parhelic angular distance from the sun, and D0 = angular distance of halo from sun.

h Ao ~ Do

0° 0° 0'

10° 0° 20'

20° 1° 14'

30° 2° 59'

40° 5° 48'

50° 10° 36'

60° 22° 48'

60° 45' 28° 14'

Arcs of Lowitz, or Vertical Arcs of the 22° Parhelia.—On rare occasions oblique extensions of the parhelia of 22°, concave towards the sun and with red inner borders, are seen, in addition to their horizontal tails, above described. These are known as the arcs of Lowitz, after the astronomer who described them2 as seen in the famous Petersburg halo complex (Fig. 194) of June 29, 1790. Their general explanation is simple, though exact computations of their outlines, and of the relative intensities of their parts for Hiffp.rp.-nt altitudes of the sun. are rather tedious. As already explaine^ are produced by ice crystals whv^

by those set to minimum refractiun, ^nu. uu.c la-ui^x uj ^±joucno UUJ.JULCVA more or less from this unique position.

When, however, the principal axes oscillate about the vertical, as they obviously do in the case of snowflakes, the arcs of Lowitz, or obliquely vertical extensions of the parhelia of 22°, necessarily are produced, though rarely seen, because of the diffusion of their light in the midst of a general glare, as explained on page 501.

Consider, first, for simplicity, the doubly special case in which the sun is on the horizon and the principal axes of the crystals oscillate in a vertical plane passing through the sun. Let C (Fig. 195) be the position of an ice crystal whose principal axis is in the direction CZ. Let an observer be at 0 and let the incident ray SC lying between Z and P make the angle h with the principal plane CP. On emerging, this ray, in its new direction S'O, has the same inclination as formerly to the axis. Hence, SC and S'C may be regarded as elements of a right cone of vertex C and .axis CZ and as the plane CZS is vertical, if S is on the horizon, as seen from 0, the element CS (lowest element), being parallel to OS, owing to the great distance of S, will lie in a plane tangent to the cone and parallel to the plane of the horizon, while every other element, such as CS', will lie above it. >S'; therefore, the apparent position of S due to

1 PERNTER-EXNER, " Meteorologische Optik," 2nd Ed., p. 367. * Nova. Ada Acad. Petropol., 8; 384, 1794.